Stephanie here from Shopify!
It's not possible to customize the checkout to create a new custom field as the checkout.liquid is highly secure.
Here are some workarounds you could use :)
You could modify the language of the checkout to change one of the fileds that is presently there to "Personal ID" by accessing the language editor (Online store > Themes > '...' > Edit Langauge > Checkout & System).
You could enable order notes which is configured within the customize theme section of your admin (Online store > Themes > Customize theme > Cart > Enable order notes).
There is also the option of customizing your cart further with cart attributes. Here's a helpful site for creating new fields for the cart - http://ui-elements-generator.myshopify.com/pages/cart-attribute
Hope these suggestions help!
We've put together an Italian tax guide for Shopify stores to help you configure your VAT (IVA or Imposta sul Valore Aggiunto) properly. We have guides for other major EU countries too, as well as non-EU countries such as Australia, Canada, India, etc.
You should also consider providing legal invoices to your Italian customers using our Shopify invoicing app for Italian stores. Our invoices display all required information, including VAT ID numbers (P.IVA or Partita Imposta sul Valore Aggiunto) of your store and business customers, as well as C.F. (Codice Fiscale) of your consumers.
Hope this helps!